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<h1>tt_canform1
</h1>

<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../../up.png"></a></h2>
<div class="box"><strong>Transforms a TT-tensor into the skeleton form</strong></div>

<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../../up.png"></a></h2>
<div class="box"><strong>function [core_int,ind_l,ind_r,ind_left,ind_right]=tt_canform1(tt) </strong></div>

<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../../up.png"></a></h2>
<div class="fragment"><pre class="comment">Transforms a TT-tensor into the skeleton form
   [CORE_INT,IND_L,IND_R]=TT_CANFORM1(TT) Transforms a TT tensor to 
   canonical skeleton-based form. The function computes left index set 
   &amp; compute right index set and also the interpolation sets


 TT-Toolbox 2.2, 2009-2012

This is TT Toolbox, written by Ivan Oseledets et al.
Institute of Numerical Mathematics, Moscow, Russia
webpage: http://spring.inm.ras.ru/osel

For all questions, bugs and suggestions please mail
ivan.oseledets@gmail.com
---------------------------</pre></div>

<!-- crossreference -->
<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../../up.png"></a></h2>
This function calls:
<ul style="list-style-image:url(../../matlabicon.gif)">
<li><a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>	Converts TT-matrix to TT1 cell-array format</li><li><a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>	Mode sizes of the TT-matrix</li><li><a href="../../tt2/@tt_tensor/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>	Converts TT-tensor TT1 to old-cell array format.</li><li><a href="../../tt2/@tt_tensor/qr.html" class="code" title="function [tt,rm]=qr(tt,op)">qr</a>	Left and right orthogonalization of the TT-format</li><li><a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>	Reshape of the TT-tensor</li><li><a href="../../tt2/@tt_tensor/size.html" class="code" title="function [sz] = size(tt,dim)">size</a>	Mode sizes of the TT-tensor</li><li><a href="../../tt2/core/maxvol2.html" class="code" title="function [ind]=maxvol2(a,ind)">maxvol2</a>	Maximal volume submatrix in an tall matrix</li><li><a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>	Tensor-by-matrix multiplication of three-dimensional tensor</li><li><a href="../../tt2/core/tt_ind2sub.html" class="code" title="function [ind] = tt_ind2sub(siz,ndx)">tt_ind2sub</a>	Correct conversion of an index to a multiindex</li></ul>
This function is called by:
<ul style="list-style-image:url(../../matlabicon.gif)">
</ul>
<!-- crossreference -->



<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../../up.png"></a></h2>
<div class="fragment"><pre>0001 <a name="_sub0" href="#_subfunctions" class="code">function [core_int,ind_l,ind_r,ind_left,ind_right]=tt_canform1(tt)</a>
0002 <span class="comment">%Transforms a TT-tensor into the skeleton form</span>
0003 <span class="comment">%   [CORE_INT,IND_L,IND_R]=TT_CANFORM1(TT) Transforms a TT tensor to</span>
0004 <span class="comment">%   canonical skeleton-based form. The function computes left index set</span>
0005 <span class="comment">%   &amp; compute right index set and also the interpolation sets</span>
0006 <span class="comment">%</span>
0007 <span class="comment">%</span>
0008 <span class="comment">% TT-Toolbox 2.2, 2009-2012</span>
0009 <span class="comment">%</span>
0010 <span class="comment">%This is TT Toolbox, written by Ivan Oseledets et al.</span>
0011 <span class="comment">%Institute of Numerical Mathematics, Moscow, Russia</span>
0012 <span class="comment">%webpage: http://spring.inm.ras.ru/osel</span>
0013 <span class="comment">%</span>
0014 <span class="comment">%For all questions, bugs and suggestions please mail</span>
0015 <span class="comment">%ivan.oseledets@gmail.com</span>
0016 <span class="comment">%---------------------------</span>
0017 tt=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>(tt);
0018 d=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(tt,1);
0019 tt1=tt;
0020 mat=tt1{1};
0021 ind_left=cell(d,1);
0022 [ind]=<a href="../../tt2/core/maxvol2.html" class="code" title="function [ind]=maxvol2(a,ind)">maxvol2</a>(mat);
0023 ind_left{1}=ind;
0024 ind_l=cell(d,1);
0025 r1=mat(ind,:);
0026 tt1{1}=mat/r1;
0027 ind_l{1}=ind;
0028 <span class="keyword">for</span> i=2:d-1
0029     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=tt1{i};
0030     ncur=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,1);
0031     r3=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3);
0032     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2,r1');
0033     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=permute(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[2,1,3]); r2=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,1);
0034     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[r2*ncur,r3]);
0035     ind=<a href="../../tt2/core/maxvol2.html" class="code" title="function [ind]=maxvol2(a,ind)">maxvol2</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>);
0036     <span class="comment">%ind(k) varies from 1 to ncur*r2 and we need to</span>
0037     <span class="comment">%convert it to two-dimensional index</span>
0038     rnew=min(ncur*r2,r3);
0039     ind_new=zeros(i,r3);
0040     ind_old=ind_left{i-1};
0041     <span class="comment">%ind_new=zeros(i,rnew);</span>
0042     <span class="comment">%ncur=sz(k);</span>
0043      <span class="keyword">for</span> s=1:rnew
0044         f_in=ind(s);
0045         w1=<a href="../../tt2/core/tt_ind2sub.html" class="code" title="function [ind] = tt_ind2sub(siz,ndx)">tt_ind2sub</a>([r2,ncur],f_in);
0046         rs=w1(1); js=w1(2);
0047         ind_new(:,s)=[ind_old(:,rs)',js];
0048      <span class="keyword">end</span>
0049     ind_left{i}=ind_new;
0050     r1=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>(ind,:);
0051     <span class="comment">%sbm_l{i}=r1;</span>
0052     ind_l{i}=ind;
0053     tt1{i}=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>/r1;
0054     rnew=min(r2*ncur,rnew);
0055     tt1{i}=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(tt1{i},[r2,ncur,rnew]);
0056     tt1{i}=permute(tt1{i},[2,1,3]);
0057 <span class="keyword">end</span>
0058 <span class="comment">%We need also right indices to do the job</span>
0059 <span class="comment">%Compute right-to-left qr &amp; maxvol</span>
0060 tt2=tt;
0061 mat=tt2{d};
0062 [q,rv]=<a href="../../tt2/@tt_tensor/qr.html" class="code" title="function [tt,rm]=qr(tt,op)">qr</a>(mat,0);
0063 tt2{d}=q;
0064 <span class="keyword">for</span> i=(d-1):-1:2
0065     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=tt2{i};
0066     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3,rv');
0067     ncur=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,1);
0068     r2=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2);
0069     r3=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3);
0070     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=permute(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[1,3,2]);
0071     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[ncur*r3,r2]);
0072     [tt2{i},rv]=<a href="../../tt2/@tt_tensor/qr.html" class="code" title="function [tt,rm]=qr(tt,op)">qr</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,0);
0073     rnew=min(r2,ncur*r3);
0074     tt2{i}=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(tt2{i},[ncur,r3,rnew]);
0075     tt2{i}=permute(tt2{i},[1,3,2]);
0076 <span class="keyword">end</span>
0077 tt2{1}=tt2{1}*rv';
0078 <span class="comment">%keyboard;</span>
0079 <span class="comment">%Now compute right-to-left maxvol</span>
0080 <span class="comment">%sbm_tmp=cell(i,1);</span>
0081 mat=tt2{d};
0082 ind_right=cell(d,1);
0083 [ind]=<a href="../../tt2/core/maxvol2.html" class="code" title="function [ind]=maxvol2(a,ind)">maxvol2</a>(mat);
0084 ind_right{d-1}=ind;
0085 r1=mat(ind,:);
0086 <span class="comment">%sbm_r=cell(d,1);</span>
0087 ind_r=cell(d,1);
0088 tt2{d}=mat/r1;
0089 ind_r{d-1}=ind;
0090 <span class="comment">%sbm_tmp{d}=r1;</span>
0091 <span class="keyword">for</span> i=(d-1):-1:2
0092     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=tt2{i};
0093     ncur=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,1);
0094     r2=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2);
0095     r3=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3);
0096     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3,r1');
0097     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=permute(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[3,1,2]);
0098     <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[r3*ncur,r2]);
0099     [ind]=<a href="../../tt2/core/maxvol2.html" class="code" title="function [ind]=maxvol2(a,ind)">maxvol2</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>); 
0100     ind_old=ind_right{i};
0101     rnew=min(ncur*r3,r2);
0102     ind_new=zeros(d-i+1,rnew);
0103     <span class="comment">%ncur=sz(k);</span>
0104      <span class="keyword">for</span> s=1:rnew
0105         f_in=ind(s);
0106         w1=<a href="../../tt2/core/tt_ind2sub.html" class="code" title="function [ind] = tt_ind2sub(siz,ndx)">tt_ind2sub</a>([r3,ncur],f_in);
0107         rs=w1(1); js=w1(2);
0108         ind_new(:,s)=[js,ind_old(:,rs)'];
0109      <span class="keyword">end</span>
0110      ind_right{i-1}=ind_new;
0111     r1=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>(ind,:);
0112     <span class="comment">%sbm_r{i-1}=r1;</span>
0113     ind_r{i-1}=ind;
0114     <span class="comment">%sbm_tmp{i}=r1;</span>
0115     tt2{i}=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>/r1;
0116     rnew=min(ncur*r3,r2);
0117     tt2{i}=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(tt2{i},[r3,ncur,rnew]);
0118     tt2{i}=permute(tt2{i},[2,3,1]);
0119 <span class="keyword">end</span>
0120   <span class="comment">%Now interpolation tensors can be computed;</span>
0121   <span class="comment">%We can look for maximal element amongst them!</span>
0122   <span class="comment">%Each int. tensor is a core convolved with sbm_l</span>
0123   <span class="comment">%from the left &amp; with smb_r from the right</span>
0124   sbm_l=cell(d-1,1);
0125 sbm_r=cell(d-1,1);
0126 sbm_l{1}=tt{1}(ind_l{1},:);
0127 <span class="keyword">for</span> k=2:d-1
0128   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=tt{k};
0129   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2,sbm_l{k-1}');
0130   ncur=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,1); r2=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2); r3=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3);
0131   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(permute(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[2,1,3]),[r2*ncur,r3]);
0132   sbm_l{k}=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>(ind_l{k},:);
0133 <span class="keyword">end</span>
0134 sbm_r{d-1}=tt{d}(ind_r{d-1},:);
0135 <span class="keyword">for</span> k=d-1:-1:2
0136   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a> = tt{k};
0137   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3,sbm_r{k}');
0138   ncur=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,1); r2=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2); r3=<a href="../../tt2/@tt_matrix/size.html" class="code" title="function [sz] = size(tt)">size</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3);
0139   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/@tt_tensor/reshape.html" class="code" title="function [tt2]=reshape(tt1,sz,eps, rl, rr)">reshape</a>(permute(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,[3,1,2]),[r3*ncur,r2]);
0140   sbm_r{k-1}=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>(ind_r{k-1},:);
0141 <span class="keyword">end</span>
0142 <span class="comment">%Finally, compute the cores (well,interpolation points are interesting</span>
0143 <span class="comment">%also)</span>
0144 core_int=cell(d,1);
0145 core_int{1}=tt{1}*sbm_r{1}';
0146 core_int{d}=tt{d}*sbm_l{d-1}';
0147 <span class="keyword">for</span> k=2:d-1
0148   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=tt{k}; <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,2,sbm_l{k-1}');
0149   <a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>=<a href="../../tt2/core/ten_conv.html" class="code" title="function [new_core] = ten_conv(core, k, mat)">ten_conv</a>(<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>,3,sbm_r{k}');
0150   core_int{k}=<a href="../../tt2/@tt_matrix/core.html" class="code" title="function [tt] = core(tt1,varargin)">core</a>;
0151 <span class="keyword">end</span>
0152 <span class="keyword">return</span>
0153 <span class="keyword">end</span></pre></div>
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